A Move Recent Review of The Integral Equations and Their
Applications

Abstract. In many branches of pure analysis, Integral Equations are one of the most useful
techniques, such as functional analysis theories and stochastic processes. It is one of the most
significant branches of mathematical analysis, in many fields of mechanics and mathematical
physics,. In this research, we will address the integral equations in many physical issues and their
applications. They are also associated with mechanical vibration problems, analytic function
theory, orthogonal systems, quadratic form theory of infinitely many variables.
1. Introduction
In physics and other applied fields, various physical problems result in initial value problems or
boundary value problems.[1],[2] and [3] Although it is equivalent to framing the problems in the form
of differential equations (ordinary and partial) or in the form of integral equations, for two main reasons,
it is preferred to choose the integral form. First the solution to the integral equation is much simpler than
the problems with the original boundary value or the initial value. The second reason is that integral
equations are better suited than differential equations for approximate methods.[4] Moreover, Integral
equations are developed for the solution of differential equations as a representation formula. With the
help of initial and boundary conditions, differential equations can be replaced by an integral equation.
As a result, the boundary conditions themselves are met by each solution of the integral
equation.Historically, Fourier (1768-1830) is the initiator of the theory of integral equations. Du Bois-
Reymond first suggested the term integral equation in 1888. Du Bois-Reymond describes an integral
equation and understands an equation in which, under one or more signs of definite integration, the
unidentified function takes place. In the late eighteenth and early ninetieth centuries, Laplace, Fourier,
Poission, Liouville and qualified studies of some basic form of integral equation. Pioneering systematic
research goes back to the work of Volterra, Fredholm and Hilbert in the late 19th and early 20th
centuries.In a paper published in 1903 in the Acta Mathematica, The basics of the Fredholm integral